Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression

• URL: http://arxiv.org/abs/2311.00140v1
• Date: Tue, 31 Oct 2023 20:25:36 GMT
• Title: Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression
• Authors: Zhaoyang Shi, Krishnakumar Balasubramanian, and Wolfgang Polonik
• Abstract summary: We show both adaptive and non-adaptive minimax rates of convergence for a family of weighted Laplacian-Eigenmap based nonparametric regression methods.
• Score: 14.003044924094597
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show both adaptive and non-adaptive minimax rates of convergence for a family of weighted Laplacian-Eigenmap based nonparametric regression methods, when the true regression function belongs to a Sobolev space and the sampling density is bounded from above and below. The adaptation methodology is based on extensions of Lepski's method and is over both the smoothness parameter ($s\in\mathbb{N}_{+}$) and the norm parameter ($M>0$) determining the constraints on the Sobolev space. Our results extend the non-adaptive result in \cite{green2021minimax}, established for a specific normalized graph Laplacian, to a wide class of weighted Laplacian matrices used in practice, including the unnormalized Laplacian and random walk Laplacian.

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